Given the coordinate matrix of x relative to a (nonstandard) basis 8 for R, find the coordinate matrix of x relative to the standard basis B-((3,-1), (0, 1)}, [x]s-

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis.

\[ B = \{(3, -1), (0, 1)\} \]

\[
[x]_B = 
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]

\[ [x]_S = 
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]

Explanation:

- The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis.
- The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \).
- The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\).
- The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.
Transcribed Image Text:Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis. \[ B = \{(3, -1), (0, 1)\} \] \[ [x]_B = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \] \[ [x]_S = \begin{bmatrix} \hspace{1cm} \\ \hspace{1cm} \end{bmatrix} \] Explanation: - The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis. - The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \). - The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\). - The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.
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